Interpolating hereditarily indecomposable Banach spaces

Argyros, S. A.¹ ; Felouzis, V.¹

¹ Department of Mathematics, University of Athens, Athens, Greece

Journal of the American Mathematical Society, Tome 13 (2000) no. 2, pp. 243-294

Résumé

The following dichotomy is proved. Every Banach space either contains a subspace isomorphic to $\ell ^1$, or it has an infinite-dimensional closed subspace which is a quotient of a Hereditarily Indecomposable (H.I.) separable Banach space. In the particular case of $L^p(\lambda ),\ 1$, it is shown that the space itself is a quotient of a H.I. space. The factorization of certain classes of operators, acting between Banach spaces, through H.I. spaces is also investigated. Among others it is shown that the identity operator $I: L^{\infty }(\lambda )\to L^1(\lambda )$ admits a factorization through a H.I. space. The same result holds for every strictly singular operator $T: \ell ^p\to \ell ^q,\ 1$. Interpolation methods and the geometric concept of thin convex sets together with the techniques concerning the construction of Hereditarily Indecomposable spaces are used to obtain the above mentioned results.

Détail

DOI : 10.1090/S0894-0347-00-00325-8

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